A049780 Array T, read by descending antidiagonals: T(n, k) = k*(2*n + k + 1)/2 for n, k >= 0.
0, 1, 0, 3, 2, 0, 6, 5, 3, 0, 10, 9, 7, 4, 0, 15, 14, 12, 9, 5, 0, 21, 20, 18, 15, 11, 6, 0, 28, 27, 25, 22, 18, 13, 7, 0, 36, 35, 33, 30, 26, 21, 15, 8, 0, 45, 44, 42, 39, 35, 30, 24, 17, 9, 0, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10, 0, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11, 0
Offset: 0
Examples
From _Petros Hadjicostas_, Nov 20 2019: (Start) Rectangular array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... 0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, ... 0, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, ... 0, 4, 9, 15, 22, 30, 39, 49, 60, 72, 85, ... 0, 5, 11, 18, 26, 35, 45, 56, 68, 81, 95, ... 0, 6, 13, 21, 30, 40, 51, 63, 76, 90, 105, ... ... (End) From _Philippe Deléham_, Mar 07 2013: (Start) Triangle S(n, k) (with rows n >= 0 and columns k >= 0) begins as follows: 0; 1, 0; 3, 2, 0; 6, 5, 3, 0; 10, 9, 7, 4, 0; 15, 14, 12, 9, 5, 0; 21, 20, 18, 15, 11, 6, 0; 28, 27, 25, 22, 18, 13, 7, 0; 36, 35, 33, 30, 26, 21, 15, 8, 0; ... (End)
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
-
GAP
Flat(List([0..15], n-> List([0..n], k-> (n-k)*(n+k+1)/2 ))); # G. C. Greubel, Dec 12 2019
-
Magma
[(n-k)*(n+k+1)/2: k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 12 2019
-
Maple
seq(seq( (n-k)*(n+k+1)/2, k=0..n), n=0..15); # G. C. Greubel, Dec 12 2019
-
Mathematica
Table[(n-k)*(n+k+1)/2, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 12 2019 *) -
PARI
T(n,k) = k*(2*n+k+1)/2; for(n=0, 15, for(k=0,n, print1(T(k,n-k), ", "))) \\ G. C. Greubel, Dec 12 2019
-
Sage
[[(n-k)*(n+k+1)/2 for k in (0..n)] for n in (0..15)] # G. C. Greubel, Dec 12 2019
Formula
T(n,k) = Sum_{j=1..k} (n+j) = k*(2*n + k + 1)/2.
S(n,k) = n*(n+1)/2 - k*(k+1)/2 for n >= 0 and 0 <= k <= n - Philippe Deléham, Mar 07 2013 [Edited by Petros Hadjicostas, Nov 20 2019]
From Stefano Spezia, Dec 13 2019: (Start)
G.f. for T(n,k): y*(1 - x*y)/((1 - x)^2*(1 - y)^3).
E.g.f. for T(n,k): (1/2)*exp(x+y)*y*(2 + 2*x + y). (End)
G.f. for S(n,k): x*(1 - x^2*y)/((1 - x*y)^2*(1 - x)^3). - Petros Hadjicostas, Dec 14 2019
Comments